We use time-resolved degenerate four-wavemixing (DFWM) with femtosecond pulses in the wavelength range 0.74-1.7 mu m to measure both phase and amplitude of all nonvanishing elements of the electronic third-order nonlinear optical susceptibility tensor c(ijkl)(-omega,omega,omega,-omega) of a 10 mu m amorphous C-60 film on a CaF2 substrate. Linear absorption is found to be less than 1% in this range. We find a single resonance in DFWM, the amplitudes and phases of which are fit well by a Lorentzian model of a two-photon resonance to a level 2.7 +/- 0.1 eV above the ground level, with width 0.25 eV. The peak two-photon absorption coefficient is 0.02 cm/MW, essentially the same peak value as for bulk gallium arsenide, one of the strongest and mast widely studied of the two-photon absorbers. Our results show there is only one two-photon allowed transition below 3.4 eV (as well as below the first one-photon transition), an unambiguous signature which is expected from theory, Theory assigns the symmetry H-g to this lowest lying two-photon state, We see a clearly nonresonant long-wavelength limit for the third-order optical susceptibility tensor which is 250 +/- 70 times the known long-wavelength limit for fused quartz (our nonlinear standard). This result is at least an order-of-magnitude larger than any of several theoretical predictions.